I. Let ω and η be 1-forms in R3 given by a) Compute dn b) Evaluate doAn
I. Let ω and η be 1-forms in R3 given by a) Compute dn b) Evaluate doAn
proof:
Let be a k-form and η be a 1-form on Rn
Let be a k-form and η be a 1-form on Rn
1. Let ω be a k-form in Rn , Π 〉 k. If k is odd, show that ωΛω 0 4. L et ω be a k-form and let ) be a /-form. Fin d d(da) Λ η_ωΛ đ7) .
1. Let ω be a k-form in Rn , Π 〉 k. If k is odd, show that ωΛω 0 4. L et ω be a k-form and let ) be a /-form. Fin d d(da) Λ η_ωΛ đ7) .
et ω be a k-form and let be a /-form. Fin
et ω be a k-form and let be a /-form. Fin
(1) Let w1, be a k-form and w2 be an l- form, both defined in an open subset UC R3. Let d : /\k (U)-ל ЛК +1 (U) be the exterior derivative of differential forms. (a) Show that d is a linear transformation of vector spaces. (b) Show that (c) Show that (d) Show that d(w) -d(d(w)) 0 for every k-form w, i.e. the map is the zero map
(1) Let w1, be a k-form and w2 be an l-...
8. Let w be the differential form yzdydz + zxdzdx + xydrdy. (i) Show that w is closed. (ii) Is w exact? If it is, find η such that dr-w. If not, explain why not.
8. Let w be the differential form yzdydz + zxdzdx + xydrdy. (i) Show that w is closed. (ii) Is w exact? If it is, find η such that dr-w. If not, explain why not.
dN dt (K-N) The change in population size (dN) per unit time (d),--, is equal to rN. What would be the population change per year (eg, dt = 1 yr) in a population of 5433 individuals, with a per capita growth rate of 0.06, and a carrying capacity of 7606?
. Show that, for every η Σ1: Τη Σ2, η(η + 1)(2n + 1) 6 k=1
Let n ez, n > 0; let do, d1,..., dn, Co,..., En be integers in the range {0, 1, 2, 3,4}. Prove: If 5*dx = 5* ex k=0 k=0 then ek = =dfor k = 0,1,...,n.
7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1
7. Let A, , An be non-empty subsets of a finite set Ω. If 1 k n and Ek is the set of elements in Ω which belong to at least k of the Ai's show that Pal i-1